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In this article, we investigate the determination of the spatial component inthe time-dependent second order coefficient of a hyperbolic equation from boththeoretical and numerical aspects. By the Carleman estimates for generalhyperbolic operators and an auxiliary Carleman estimate, we establish localH\"older stability with both partial boundary and interior measurements undercertain geometrical conditions. For numerical reconstruction, we minimize aTikhonov functional which penalizes the gradient of the unknown function. Basedon the resulting variational equation, we design an iteration method which isupdated by solving a Poisson equation at each step. One-dimensional prototypeexamples illustrate the numerical performance of the proposed iteration.

Theoretical stability in coefficient inverse problems for general hyperbolic equations with numerical reconstruction